Geometry & Topology
- Geom. Topol.
- Volume 19, Number 3 (2015), 1249-1262.
Quasigeodesic flows and sphere-filling curves
Given a closed hyperbolic –manifold with a quasigeodesic flow, we construct a –equivariant sphere-filling curve in the boundary of hyperbolic space. Specifically, we show that any complete transversal to the lifted flow on has a natural compactification to a closed disc that inherits a –action. The embedding extends continuously to the compactification, and restricts to a surjective –equivariant map on the boundary. This generalizes the Cannon–Thurston theorem, which produces such group-invariant space-filling curves for fibered hyperbolic –manifolds.
Geom. Topol., Volume 19, Number 3 (2015), 1249-1262.
Received: 22 July 2013
Revised: 25 February 2014
Accepted: 26 July 2014
First available in Project Euclid: 16 November 2017
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Frankel, Steven. Quasigeodesic flows and sphere-filling curves. Geom. Topol. 19 (2015), no. 3, 1249--1262. doi:10.2140/gt.2015.19.1249. https://projecteuclid.org/euclid.gt/1510858761